Characterizing compact Clifford semigroups that embed into convolution and functor-semigroups

TL;DR

This paper characterizes when compact Clifford semigroups can embed into convolution and functor-semigroups, linking algebraic, topological, and categorical properties with embeddings into specific semigroups.

Contribution

It provides necessary and sufficient conditions for embedding compact Clifford semigroups into convolution and functor-semigroups, connecting algebraic structure with topological and categorical properties.

Findings

Embedding into $P(G)$ occurs iff embedding into $ ext{exp}(G)$ and the semigroup is inverse with zero-dimensional maximal semilattice.

Such semigroups embed into $F(G)$ for suitable $G$ and functors $F$ with specific properties.

Characterizes the algebraic and topological structure needed for embeddings into convolution and functor-semigroups.

Abstract

We study algebraic and topological properties of the convolution semigroups of probability measures on a topological groups and show that a compact Clifford topological semigroup $S$ embeds into the convolution semigroup $P(G)$ over some topological group $G$ if and only if $S$ embeds into the semigroup $\exp(G)$ of compact subsets of $G$ if and only if $S$ is an inverse semigroup and has zero-dimensional maximal semilattice. We also show that such a Clifford semigroup $S$ embeds into the functor-semigroup $F(G)$ over a suitable compact topological group $G$ for each weakly normal monadic functor $F$ in the category of compacta such that $F(G)$ contains a $G$-invariant element (which is an analogue of the Haar measure on $G$).